Dissertations / Theses on the topic 'Measure Theoretic Probability Theory'

Pathfinding, as a process of selecting a fixed route, has long been studied in Computer Science and Mathematics. Decision making, as a similar, but intrinsically different, process of determining a control policy, is much less studied. Here, I propose a problem that appears to be of the first class, which would suggest that it is easily solvable with a modern machine, but that would be too easy, it turns out. By allowing a pathfinding to anticipate and respond to information, without setting restrictions on the \structure" of this anticipation, selecting the \best step" appears to be an intractable problem. After introducing the necessary foundations and stepping through the strangeness of “safest-with-sight," I attempt to develop an method of approximating the success rate associated with each potential decision; the results suggest something fundamental about decision making itself, that information that is collected at a moment that it is not immediately “consumable", i.e. non-incident, is not as necessary to anticipate than the contrary, i.e. incident information. This is significant because (i) it speaks about when the information should be anticipated, a moment in decision-making long before the information is actually collected, and (ii) whenever the model is restricted to only incident anticipation the problem again becomes tractable. When we only anticipate what is most important, solutions become easy to compute, but attempting to anticipate any more than that and solutions may become impossible to find on any realistic machine.

Thesis (Ph. D.)--University of California, Riverside, 2009.
Includes abstract. Also issued in print. Includes bibliographical references (leaves 81-84). Available via ProQuest Digital Dissertations.

Let μ be a Radon probability measure on M, the d-dimensional Real Euclidean space (where d is a positive integer), and f a measurable function. Let P be the space of sequences whose coordinates are elements in M. Then, for any point x in M, define a function ƒn on M and P that looks at the first n terms of an element of P and evaluates f at the first of those n terms that minimizes the distance to x in M. The measures for which such sequences converge in measure to f for almost every sequence are called Mycielski-regular. We show that the self-similar measure generated by a finite family of contracting similitudes and which up to a constant is the Hausdorff measure in its dimension on an invariant set C is Mycielski-regular.

The ideals of exact modeling, and of putting off approximations as long as possible, make Bayesian practice both successful and difficult. Languages for modeling probabilistic processes, whose implementations answer questions about them under asserted conditions, promise to ease much of the difficulty. Unfortunately, very few of these languages have mathematical specifications. This makes them difficult to trust: there is no way to distinguish between an implementation error and a feature, and there is no standard by which to prove optimizations correct. Further, because the languages are based on the incomplete theories of probability typically used in Bayesian practice, they place seemingly artificial restrictions on legal programs and questions, such as disallowing unbounded recursion and allowing only simple equality conditions. We prove it is possible to make trustworthy probabilistic languages for Bayesian practice by using functional programming theory to define them mathematically and prove them correct. The specifications interpret programs using measure-theoretic probability, which is a complete enough theory of probability that we do not need to restrict programs or conditions. We demonstrate that these trustworthy languages are useful by implementing them, and using them to model and answer questions about typical probabilistic processes. We also model and answer questions about processes that are either difficult or impossible to reason about precisely using typical Bayesian mathematical tools.

In this thesis, we will develop the fundamental properties of financial mathematics, with a focus on establishing meaningful connections between martingale theory, stochastic calculus, and measure-theoretic probability. We first consider a simple binomial model in discrete time, and assume the impossibility of earning a riskless profit, known as arbitrage. Under this no-arbitrage assumption alone, we stumble upon a strange new probability measure Q, according to which every risky asset is expected to grow as though it were a bond. As it turns out, this measure Q also gives the arbitrage-free pricing formula for every asset on our market. In considering a slightly more complicated model over a finite probability space, we see that Q once again makes its appearance. Finally, in the context of continuous time, we build a framework of stochastic calculus to model the trajectories of asset prices on a finite time interval. Under the absence of arbitrage once more, we see that Q makes its return as a Radon-Nikodym derivative of our initial probability measure. Finally, we use the properties of Q and a stochastic differential equation that models the dynamics of the assets of our market, known as the Ito formula, in order to derive the classic Black-Scholes Equation.

With the current proliferation of cyber attacks, safeguarding internet facing assets from network intrusions, is becoming a vital task in our increasingly digitalised economies. Although recent successes of machine learning (ML) models bode the dawn of a new generation of intrusion detection systems (IDS); current solutions struggle to implement these in an efficient manner, leaving many IDSs to rely on rule-based techniques. In this paper we begin by reviewing the different approaches to feature construction and attack source identification employed in such applications. We refer to these steps as the framework within which models are implemented, and use it as a prism through which we can identify the challenges different solutions face, when applied in modern network traffic conditions. Specifically, we discuss how the most popular framework -- the so called flow-based approach -- suffers from significant overhead being introduced by its resource heavy pre-processing step. To address these issues, we propose the Information Theoretic Framework for Network Anomaly Detection (ITF-NAD); whose purpose is to facilitate online application of statistical learning models onto high-speed network links, as well as provide a method of identifying the sources of traffic anomalies. Its development was inspired by previous work on information theoretic-based anomaly and outlier detection, and employs modern techniques of entropy estimation over data streams. Furthermore, a case study of the framework's detection performance over 5 different types of Denial of Service (DoS) attacks is undertaken, in order to illustrate its potential use for intrusion detection and mitigation. The case study resulted in state-of-the-art performance for time-anomaly detection of single source as well as distributed attacks, and show promising results regarding its ability to identify underlying sources.
I takt med att antalet cyberattacker växer snabbt blir det alltmer viktigt för våra digitaliserade ekonomier att skydda uppkopplade verksamheter från nätverksintrång. Maskininlärning (ML) porträtteras som ett kraftfullt alternativ till konventionella regelbaserade lösningar och dess anmärkningsvärda framgångar bådar för en ny generation detekteringssytem mot intrång (IDS). Trots denna utveckling, bygger många IDS:er fortfarande på signaturbaserade metoder, vilket förklaras av de stora svagheter som präglar många ML-baserade lösningar. I detta arbete utgår vi från en granskning av nuvarande forskning kring tillämpningen av ML för intrångsdetektering, med fokus på de nödvändiga steg som omger modellernas implementation inom IDS. Genom att sätta upp ett ramverk för hur variabler konstrueras och identifiering av attackkällor (ASI) utförs i olika lösningar, kan vi identifiera de flaskhalsar och begränsningar som förhindrar deras praktiska implementation. Särskild vikt läggs vid analysen av de populära flödesbaserade modellerna, vars resurskrävande bearbetning av rådata leder till signifikant tidsfördröjning, vilket omöjliggör deras användning i realtidssystem. För att bemöta dessa svagheter föreslår vi ett nytt ramverk -- det informationsteoretiska ramverket för detektering av nätverksanomalier (ITF-NAD) -- vars syfte är att möjliggöra direktanslutning av ML-modeller över nätverkslänkar med höghastighetstrafik, samt tillhandahåller en metod för identifiering av de bakomliggande källorna till attacken. Ramverket bygger på modern entropiestimeringsteknik, designad för att tillämpas över dataströmmar, samt en ASI-metod inspirerad av entropibaserad detektering av avvikande punkter i kategoriska rum. Utöver detta presenteras en studie av ramverkets prestanda över verklig internettrafik, vilken innehåller 5 olika typer av överbelastningsattacker (DoS) genererad från populära DDoS-verktyg, vilket i sin tur illustrerar ramverkets användning med en enkel semi-övervakad ML-modell. Resultaten visar på hög nivå av noggrannhet för detektion av samtliga attacktyper samt lovande prestanda gällande ramverkets förmåga att identifiera de bakomliggande aktörerna.

In this thesis, the performance of two over-sampling techniques, SMOTE and ADASYN, is compared. The comparison is done on three imbalanced data sets using three different classification models and evaluation metrics, while varying the way the data is pre-processed. The results show that both SMOTE and ADASYN improve the performance of the classifiers in most cases. It is also found that SVM in conjunction with SMOTE performs better than with ADASYN as the degree of class imbalance increases. Furthermore, both SMOTE and ADASYN increase the relative performance of the Random forest as the degree of class imbalance grows. However, no pre-processing method consistently outperforms the other in its contribution to better performance as the degree of class imbalance varies.

We introduce a simple tree growth process that gives rise to a new two-parameter family of discrete fragmentation trees that extends Ford's alpha model to multifurcating trees and includes the trees obtained by uniform sampling from Duquesne and Le Gall's stable continuum random tree. We call these new trees the alpha-gamma trees. In this thesis, we obtain their splitting rules, dislocation measures both in ranked order and in sized-biased order, and we study their limiting behaviour. We further extend the underlying exchangeable fragmentation processes of such trees into partly exchangeable fragmentation processes by weakening the exchangeability. We obtain the integral representations for the measures associated with partly exchangeable fragmentation processes and subordinator of the tagged fragments. We also embed the trees associated with such processes into continuum random trees and study their limiting behaviour. In the end, we generate a three-parameter family of partly exchangeable trees which contains the family of the alpha-gamma trees and another important two-parameter family based on Poisson-Dirichlet distributions.

Context: Medical management for head and neck cancer (HNC) often includes neck dissection surgery, a side effect of which is shoulder dysfunction. There is no consensus for which patient-reported outcome measure (PRO) is most appropriate to quantify shoulder dysfunction in this population. Objective: The aims of this research study were to: (1) use Rasch methodologies to assess construct validity and overall appropriateness of test score interpretation of Disability of the Arm, Shoulder and Hand (DASH), QuickDASH, Shoulder Pain and Disability Index (SPADI) and Neck Dissection Impairment Index (NDII) in the HNC population; (2) determine appropriateness of use of University of Washington Quality of Life (UW-QoL) shoulder subscale as a screening tool for shoulder impairment; (3) recommend a new PRO, or combination of PROs, that more accurately portrays the construct of shoulder dysfunction in the HNC population. Design: One hundred and eight-two individuals who had received a neck dissection procedure within the past 2 weeks to 18 months completed the PROs. Rasch methodologies were utilized to address the primary aim of the study through consideration of scale dimensionality [principal components analysis, item and person fit, differential item functioning (DIF)], scale hierarchy (gaps/redundancies, floor/ceiling effects, coverage of ability levels), response scale structure, and reliability (person and item reliability and separation statistics). The secondary aim was addressed through correlational analysis of the UW-QoL (shoulder subscale), DASH, QuickDASH, SPADI and NDII. Results: The DASH did not meet criteria for unidimensionality, and was deemed inappropriate for utilization in this sample. The QuickDASH, SPADI and NDII were all determined to be unidimensional. All scales had varying issues with person and item misfit, DIF, coverage of ability levels, gaps/redundancies, and optimal rating scale requirements. The NDII meets most requirements. All measures were found to meet thresholds for person and item separation and reliability statistics. The third aim of this study was not addressed because the NDII was determined to be appropriate for this population. Conclusions: Rasch analysis indicates the NDII is the most appropriate measure studied for this population. The QuickDASH and SPADI are recommended with reservation. The DASH and the UW-QoL (shoulder subscale) are not recommended.

L'objectif de cette thèse est d'étudier le comportement local d'une mesure de probabilité, notamment au travers d'un indice de régularité locale. Dans la première partie, nous établissons la normalité asymptotique de l'estimateur des kn plus proches voisins de la densité et de l'histogramme. Dans la deuxième, nous définissons un estimateur du mode sous des hypothèses affaiblies. Nous montrons que l'indice de régularité intervient dans ces deux problèmes. Enfin, nous construisons dans une troisième partie différents estimateurs pour l'indice de régularité à partir d'estimateurs de la fonction de répartition, dont nous réalisons une revue bibliographique.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1993.
Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 86-88).

在這篇論文中，我們將會首先討論什麼概率測度μ 上的L²空間會存在指數型正交基(exponential orthonormal basis) ，而μ 則稱為一個譜測度若指數型正交基存在。這個問題源於1974年的Fuglede猜想和Jorgensen與Pedersen對分形譜測度存在性的研究。我們有興趣理解怎麼樣的測度會是譜測度，而對於沒有指數型正交基的測度，我們則探討它們會否存在更廣義且在Fourier分析中常用的指數型基，如Riesz基或Fourier框架(Fourier frame) 。
我們知道一個測度可以唯一分解成離散、奇異和絶對連續三部份。我們首先証明譜測度肯定是純型(pure type) 。若測度是絶對連續，我們對有Fourier框架的測度的密度給出一個完全的刻直到。這個結果對研究Gabor框架的問題都有幫助。對於離散且只有有限個非零質量原子的測度，我們証明它們全都都有Riesz基。在最困難作出一般討論的奇異測度中，我們透過譜測度與離散測度的卷積找出了有Riesz基但沒有指數型正交基的奇異測度。我們進而探討自彷函數送代系統(affine IFSs) ，我們証明到如果一個自彷函數送代系統是測度分離且有Fourier框架，那麼它在每一個函數的概率權都是一樣的。我們亦証明了Laba-Wang猜想在絶對連續的自相似測度上是正確的。這些結果都表示了一個有Fourier框架的測度都應該在其支撐上有一定的均勻性。
在論文的第二部分我們會探討自彷tile其Fourier變換的零點集。在自彷tile的研究中，其中一個基本問題就是刻劃其數字集(digit set)使得那自彷函數送代系統的不變集能以平移密鋪空間。透過Kenyon條件，我們可將這個問題轉化成理解Fourier變換的零點集。男一方面，指數型正交基的存在性亦需要我們探討Fourier變換的零點集，而自彷tile 的Fourier變換是可以明確寫出來的。這使自彷tile成為一個很好去研究tilings和譜測度相互關係的好例子。
我們利用了分圓多項式(cyclotomic polynomials)對一維自彷tile的零點集進行了一個詳細的研究。從這裡我們把tile的數字集寫成某些分圓多項式的乘積。這個乘積亦可以一個樹上的切集(blocking)表示出來。這個表示亦把tile數字集的乘積形式(product-forms) 一般化成高階乘積形式。我們証明了在任何維數的tile數字集都是整數tile(即它們能平移密鋪整數集Z) 。這個結果讓我們能使用Coven和Meyerowitz所提出的整數tile分解方法，來使tile數字集完整刻劃成高階模乘積形式(high order modulo product-forms) 當數字集的數目為p[superscript α]q而p，q則是質數。由於我們對零點集亦完全清楚，這對在自彷tile上尋找完備的指數型正交基提供了一個新的方向。
In this thesis, we will first consider when a probability measure μ admits an exponential orthonormal basis on its L² space (μ is called spectral measures).This problem originates from the conjecture of Fuglede in 1974, and the discovery of Jorgensen and Pedersen that some fractal measures also admit exponential orthonormal bases, but some do not. It generates a lot of interest in understand- ing what kind of measures are spectral measures. For those measures failing to have exponential orthonormal bases, it is interesting to know whether such mea- sures still have Riesz bases and Fourier frames, which are generalized concepts of orthonormal bases with wide range of uses in Fourier analysis.
It is well-known that a measure has a unique decomposition as the discrete, singular and absolutely continuous parts. We first show that spectral measures must be of pure type. If the measure is absolutely continuous, we completely classify the class of densities of the measures with Fourier frames. This result has new applications to topics in applied harmonic analysis, like the Gabor analysis. For the discrete measures with finite number of atoms, we show that they all have Riesz bases. For the case of singular measure, which is the most difficult one, we show that there exist measures with Riesz bases but not orthonormal bases by considering convolution between spectral measures and discrete measures. We then investigate affine iterated function systems (IFSs), we show that if an IFS has measure disjoint condition and admits a Fourier frame, then the probability weights are all equal. Moreover, we also show that the Łaba-Wang conjecture is true if the self-similar measure is absolutely continuous. These results indicate that measures with Fourier frames must have certain kind of uniformity on the support.
In the second part of the thesis we study the zero sets of Fourier transform of self-affine tiles. One of the fundamental problems in self-affine tiles is to classify the digit sets so that the attractors form tiles. This problem can be turned to study the zeros of the Fourier transform via the Kenyon criterion. On the other hand, existence of exponential orthonormal bases requires us to know the zero sets of the Fourier transform. Self-affine tiles are translational tiles arising from IFSs with its Fourier transform written explicitly. It therefore serves as an ideal place to investigate the relation of tilings and spectral measures.
We carry out a detail study in the zero sets of the one-dimensional tiles using cyclotomic polynomials. From this we characterize the tile digit sets through some product of cyclotomic polynomials represented in terms of a blocking in a tree, which is a generalization of the product-form to higher order. We show that tile digit sets in any dimension are integer tiles. This result allows us to use the decomposition method of integer tiles by Coven and Meyerowitz to provide the explicit classification of the tile digit sets in terms of the higher order modulo product-forms when number of the digits is p[superscript α]q, p, q are primes. Since the zero sets are completely known, this provides us a new approach to study the existence of complete orthogonal exponentials in the self-affine tiles on R¹.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.
Lai, Chun Kit.
Thesis (Ph.D.)--Chinese University of Hong Kong, 2012.
Includes bibliographical references (leaves 128-135).
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Abstract also in Chinese.
Chapter 1 --- Introduction --- p.9
Chapter 1.1 --- Background and Motivations --- p.9
Chapter 1.2 --- Results on spectral measures --- p.13
Chapter 1.3 --- Results on self-affine tiles --- p.16
Chapter 2 --- Fourier Frames: Absolutely Continuous Measures --- p.21
Chapter 2.1 --- Beurling densities --- p.22
Chapter 2.2 --- Law of pure type --- p.28
Chapter 2.3 --- Absolutely continuous F-spectral measures --- p.31
Chapter 2.3.1 --- Proof by Beurling densities --- p.32
Chapter 2.3.2 --- Proof by translational absolute continuity --- p.35
Chapter 2.4 --- Applications to applied harmonic analysis --- p.40
Chapter 2.5 --- Remarks and open questions --- p.42
Chapter 3 --- Fourier Frames: Discrete and Singular Measures --- p.45
Chapter 3.1 --- Discrete measures --- p.46
Chapter 3.2 --- Convolutions with discrete measures --- p.50
Chapter 3.3 --- Self-affine measures --- p.56
Chapter 3.4 --- Iterated function systems on R¹ --- p.65
Chapter 3.5 --- Concluding remarks --- p.70
Chapter 4 --- Spectral structure of tile digit sets --- p.74
Chapter 4.1 --- Preliminaries --- p.76
Chapter 4.2 --- Modulo product-forms --- p.81
Chapter 4.3 --- Higher order product-forms --- p.86
Chapter 4.4 --- Φ-tree, blocking and kernel polynomials --- p.90
Chapter 5 --- Classifications of tile digit sets --- p.101
Chapter 5.1 --- Tile digit sets --- p.101
Chapter 5.2 --- The p[superscript α]q[superscript β] integer tiles --- p.105
Chapter 5.3 --- Tile digit sets for b = p[superscript α]q --- p.112
Chapter 5.4 --- Self-similar measures: Absolute continuity --- p.122
Chapter 5.5 --- Remarks --- p.126

In this thesis we shall examine the role of measurability in the theory of Lebesgue Integration. This shall be done in the context of the real line where we define the notion of an integral of a bouuded real-valued function over a set of bounded outer measure without a prior assumption of measurability concerning the function and the domain of integration

While decision-making under uncertainty is a major universal problem, its implications in the field of transportation systems are especially enormous; where the benefits of right decisions are tremendous, the consequences of wrong ones are potentially disastrous. In the realm of highway systems, decisions related to the highway configuration (number of lanes, right of way, etc.) need to incorporate both the traffic demand and land price uncertainties. In the literature, these uncertainties have generally been modeled using the Geometric Brownian Motion (GBM) process, which has been used extensively in modeling many other real life phenomena. But few scholars, including those who used the GBM in highway configuration decisions, have offered any rigorous justification for the use of this model. This thesis attempts to offer a detailed analysis of various aspects of transportation systems in relation to decision-making. It reveals some general insights as well as a new concept that extends the notion of opportunity cost to situations where wrong decisions could be made. Claiming deficiency of the GBM model, it also introduces a new formulation that utilizes a large and flexible parametric family of jump models (i.e., Lévy processes). To validate this claim, data related to traffic demand and land prices were collected and analyzed to reveal that their distributions, heavy-tailed and asymmetric, do not match well with the GBM model. As a remedy, this research used the Merton, Kou, and negative inverse Gaussian Lévy processes as possible alternatives. Though the results show indifference in relation to final decisions among the models, mathematically, they improve the precision of uncertainty models and the decision-making process. This furthers the quest for optimality in highway projects and beyond.